Problem: Rewrite as a simplified fraction. ${1.2\overline{1} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 121.1111...\\ 10x &= 12.1111...\end{align*}}$ Subtracting the two equations, we have: ${90x = 109}$ ${x = \dfrac{109}{90}} \text{or} {1 \dfrac{19}{90}}$